Black Holes in String Theory - calculation help needed!

I am having trouble using equations (2.1) and (2.4) to derive (2.5) and (2.6). When I do the calculation, I do not get any [itex]\phi'[/itex] terms (i.e. first derivatives --- I do get all the second-derivatives of phi and all other terms).

For example, I don't understand how the (tt) and (θθ) equations can have [itex]\phi'[/itex] terms. Doesn't (2.1a) contain
[itex]\nabla_{a}\nabla_{b}\phi = \partial_a \partial_b \varphi(r) = 0[/itex] unless [itex]a = b = r [/itex]?
Thus shouldn't only the (rr) equation contain a [itex]\phi[/itex]-term... and shouldn't it just be a second derivative?

For another example, since [itex]\nabla \phi[/itex] is first-order in [itex]\lambda[/itex], isn't [itex](\nabla \phi)^2[/itex] second-order in [itex]\lambda[/itex] and thus irrelevant for equation (2.6)? This would lead me to conclude that there should be no first derivative of [itex]\phi[/itex] in (2.6). Where does it come from?

I would greatly appreciate some insight. Tell me where I am going wrong!

You are right - I have accidentally been using [itex]\phi'[/itex] and [itex]\varphi'[/itex] interchangeably, since they differ only by a multiplicative factor. You can see this from equation (2.4) since [itex]\phi_0[/itex] is an arbitrary constant.

But I still don't understand why [itex]\varphi'[/itex] should appear anywhere in any equation of (2.5).

I can rephrase the question as:

Using (2.4), what is [itex]\nabla_a \nabla_b \phi[/itex] ?

It seems like a very easy question to me, but my answer to that questions seems to be in contradiction with what is in (2.5).

Staff: Mentor

This equality is not correct in curved spacetime, and it looks like the text is using curved spacetime (i.e., gravity is present). In curved spacetime there are extra terms in the covariant derivative [itex]\nabla_a[/itex] involving the connection coefficients.

Thank you very much! You figured it out. Indeed [itex]\nabla_a \phi = \partial_a \phi[/itex] since [itex]\phi[/itex] is a scalar field, but the second covariant derivative must incorporate the connection coefficients and the [itex]\partial_a \phi[/itex]'s. I feel silly for overlooking this - my mind was stuck for very long.